Lesson 3 – Standard Scores

/

• Introduction to Standard (z) Score

  • The standard score, also known as the z-score, is the deviation of a score from the distributed mean of the scores.
  • The z-score indicates how far a score is from the mean in standard deviation units.
  • A positive z-score means the score is greater than the mean, while a negative z-score means the score is less than the mean.
  • Standard scores allow comparison of scores between different datasets, such as in psychological tests.
  • The mean of z-scores is always zero, and the standard deviation is one.

➥ Properties of z-scores

Key characteristics:

  • The mean of z-scores is 0.
  • The standard deviation of z-scores is 1.
  • A positive z-score indicates the score is above the mean.
  • A negative z-score indicates the score is below the mean.
  • The shape of the z-score distribution is the same as the original distribution.
  • The sum of squared z-scores equals the number of z-score values.

◦ Advantages:

  • Converting raw scores to z-scores doesn’t change the distribution’s characteristics.
  • Z-scores allow comparison of scores from different distributions.

◦ Disadvantages:

  • Positive and negative signs can be confusing.
  • Decimals can be confusing.

• Transforming Raw Scores into z-scores

To convert raw scores to z-scores:

  • Calculate the mean and standard deviation of the distribution.
  • Use the formula: Z = (X – M) / σ, where X is the raw score, M is the mean, and σ is the standard deviation.

◦ Example:

  • Rita scored 70 in Section A (Mean = 50, SD = 10).
  • Surabhi scored 80 in Section B (Mean = 70, SD = 20).
  • Rita’s z-score: Z = (70 – 50) / 10 = 2
  • Surabhi’s z-score: Z = (80 – 70) / 20 = 0.5
  • Comparison: Rita performed better relative to her class.

• Determining Raw Scores from z-scores

To convert z-scores back to raw scores:

  • Use the formula: X = Zσ + M, where X is the raw score, Z is the z-score, σ is the standard deviation, and M is the mean.

◦ Example:

A student has a z-score of 1.5 in a test with a mean of 60 and a standard deviation of 8.

Raw score: X = 1.5 * 8 + 60 = 72

• Some Common Standard Scores

Besides z-scores, there are other standard scores used in psychology.

➥ T-Scores

  • T-scores convert z-scores to a distribution with a mean of 50 and a standard deviation of 10.
  • Formula: T = 10Z + 50
  • T-scores eliminate negative numbers and decimals.
  • Example: A student with a z-score of -2 would have a T-score of 30.

➥ Stanine-Score

  • Stanine (standard nine) scores divide a distribution into nine categories.
  • The mean of stanine scores is 5, and the standard deviation is approximately 2.
  • Stanines are always positive whole numbers.

➥ STEN-Score

  • STEN (standard ten) scores divide a distribution into ten categories.
  • STEN scores have a mean of 5.5 and a standard deviation of 2.
  • STEN scores are also positive whole numbers.
  • Both Stanine and STEN scores simplify comparisons but lose some precision.

• Computations of Percentiles and Percentile Ranks from Grouped Data

  • Percentiles indicate the point in a distribution below which a given percentage of scores fall.
  • Percentiles divide a dataset into 100 equal parts. Each percentile point indicates the value below which a certain percentage of observations fall.

P₁₀ → 10th percentile

P₂₅ → 25th percentile (also called 1st quartile, Q₁)

P₅₀ → 50th percentile (median)

P₇₅ → 75th percentile (3rd quartile, Q₃)

P₉₀ → 90th percentile

➥ To calculate a percentile (e.g., the 20th percentile, P20) from grouped data:

1.Identify the class interval where the percentile falls.

  • Find the score below which the desired percentage of scores fall.
  • Calculate the percentile’s position (e.g., 20% of the total number of scores).
  • Locate the interval containing that position.

2.Determine how far into the interval the percentile lies.

  • Find the number of cases from the lower real limit of the interval up to the percentile position.

3.Calculate the exact percentile value.

  • Assume scores are evenly distributed within the interval.
  • Use the formula to find the precise point within the interval: Pn = L + ((nN/100 – F) / f) × i

Breakdown of formula: –

  • Pn = Desired percentile (e.g., P₃₀, P₆₀)
  • L = Lower boundary of the percentile class
  • N = Total number of scores
  • n = Desired percentile (e.g., 30 for P₃₀)
  • F = Cumulative frequency before the percentile class
  • f = Frequency of the percentile class
  • i = Class interval size.

Step by step procedure 

  1. Calculate  nN/100 to determine the position of the desired percentile.
  2. Identify the class interval where that position falls (this is the percentile class).
  3. Use the formula to compute Pn.

• Calculating Percentile Ranks from Grouped Data

  • Percentile ranks tell you the percentage of scores in a distribution that are below a particular score.
  • For instance, if a student’s score is at the 75th percentile rank, it means that 75% of the students scored lower than that student.

Formula to calculate percentile rank: P = (n/N) * 100.

◦ Breakdown of formula: –

P = Percentile Rank

n = The number of scores below the score of interest

N = The total number of scores

◦ Steps involved in calculation

  • Arrange the data into a grouped frequency distribution.
  • Identify the raw score for which you want to find the percentile rank.
  • Count the number of scores below the interval containing the raw score.
  • Use the formula to calculate the percentile rank.

•  Comparison of Z-scores and percentile Ranks.

Both help in understanding where a score falls a set of data, but they do so in different ways.

➥ Z-Scores:

  • Express a score’s position in terms of standard deviations from the mean.
  • A z-score of 0 means the score is equal to the mean.
  • A positive z-score means the score is above the mean, and a negative z-score means it’s below the mean.

➥ Percentile Ranks:

  • Indicate the percentage of scores that are lower than a specific score.
  • They range from 0 to 100.
  •  A percentile rank of 70 means that 70% of the scores are below that particular score.

➥ Key Differences in What They Show:

  • Z-scores use standard deviation units to show how far a score is from the average.
  • Percentile ranks use percentages to show where a score stands relative to other scores in the distribution.

•  Comparison of Z-scores and percentile Ranks.

The normal distribution is a theoretical distribution commonly used in statistics.

➥ Nature of the Normal Distribution

  • It models many real-world phenomena (e.g., heights, IQ scores).
  • It’s crucial for statistical inference.

➥ Properties of the Normal Distribution

Defined by mean (µ) and standard deviation (σ).

Properties:

  1. Symmetric around the mean.
  2. Mean, median, and mode are equal.
  3. The total area under the curve is 1.
  4. Tails extend infinitely.
  5. Shape determined by mean and standard deviation.

➥ Applications of the Normal Distribution

Distribution is widely used in statistics and scientific research and is applied in the following fields:

  1. Quality control 
  2. Inferential Statistics 
  3. Financial modeling 
  4. Epidemiology 
  5. Psychology 

➥ Normal Curve and Standard Scores

  • The normal curve is a graphical representation of the normal distribution.
  • Standard scores (z-scores) can be plotted on the normal curve.

• Finding Areas when the Score is known and Finding Scores when the Area is known

  • Z-scores and the normal distribution table can be used to find the area of (proportion of scores) above or below a given score.
  • Conversely, you can find the Z-scores corresponding to a given area.

Example

1.Finding Area (Proportion) When You Know the Score:

  • Let’s say the average score is 70, and the standard deviation is 10.
  • A student scored 80.
  • We can calculate the z-score for 80: z = (80 – 70) / 10 = This means the student’s score is 1 standard deviation above the mean.
  • Using a z-table or calculator, we can find the area under the curve beyond this z-score. This area tells us the proportion of students who scored higher than 80.

2.Finding the Score When You Know the Area (Proportion):

  • Suppose we want to find the score that separates the top 10% of students.
  • We would look up the z-score that corresponds to the top 10% area in the z-table.
  • Then, we’d use the z-score formula to convert that z-score back into a raw score. This would tell us the minimum score needed to be in the top 10%.

*T-score: Was first used by William A. McCall. T of T-score is given to honour the renowned physiologists, Terman and Thorndike.